What is Differential equations: Definition and 999 Discussions

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

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  1. D

    Generalized eigenvectors and differential equations

    Let A be an 3x3 matrix such that A\mathbf{v_1}=\mathbf{v_1}+\mathbf{v_2}, A\mathbf{v_2}=\mathbf{v_2}+\mathbf{v_3}, A\mathbf{v_3}=\mathbf{v_3} where \mathbf{v_3} \neq \mathbb{0}. Let B=S^{-1}AS where S is another 3x3 matrix. (i) Find the general solution of \dot{\mathbf{x}}=B\mathbf{x}. (ii)...
  2. T

    Solving System of Two Differential Equations

    Homework Statement Find General Solution of the Following System (2D+5)x - (2D+3)y = t (D-2)x + (D+2)y = 0 https://dl.dropboxusercontent.com/u/32294083/Emath/New%20Doc%203_1.jpg Using the Quadratic Formula I get nothing so I am not sure what the complementary solution is...
  3. I

    Dividing differential equations

    The point of my question is that when we divide a differential equation by a function or variable we result in different solution (not always). Take the example: ydx+ydy=0, constaint: xy=a By substituting x with y/a and after some manipulations we arrive to (-a/y)dy+ydy=0 and on...
  4. F

    Differential equations incongruecy

    Homework Statement I am going to copy-paste this text that my friend made (because we both have the same doubt and we don't know to work around it. This is a long post, so warning): "I'm currently unsure of how these two problems work. I've tried working at them in different ways but i don't...
  5. F

    Radioactive samples and differential equations

    Homework Statement A certain nuclear plant produces radioactive waste in the form of strontium-90 at the constant rate of 500kg. The waste decays exponentially with a half-life of 28 years. How much of this radioactive waste from the nuclear plant will be present after the following...
  6. F

    Differential equations and law of cooling

    Homework Statement A homicide victim is found to have a temperature of 31°C at the stroke of midnight. At 1:00AM his temperature dropped to 29°C. Assuming that the temperature of the room stays at 20°C, when did the murder take place? Homework Equations - The Attempt at a Solution...
  7. F

    FO Differential equations and account balance

    Homework Statement a. Assume that yo dollars are deposited into an account paying r percent compounded continuously. If withdrawals are at an annual rate of 200t dollars (assume these are continuous), find the amount in the account after T years. b. Consider the special case if r = 10% and...
  8. J

    Suggestions for Differential Equations Text

    As a high school student, I enjoy studying mathematics on my spare time. Having recently worked through a textbook on vector calculus, I am in need of a textbook that will give me a solid introduction to differential equations. Any suggestions will help my search; however, I would prefer a...
  9. G

    Need a quick favor if anyone has Differential Equations 3rd Ed by Zill

    Hi Fellows: If anyone has access to a copy of 'Differential Equations with Boundary-Value Problems (THIRD edition)' by Dennis G. Zill and Michael R. Cullen, I just need you to paste here or dictate to me on phone the following problem equations: 1. Problem no. 12 of chapter 4 review...
  10. N

    Numerical solution of partial differential equations

    can anyone direct me to a website that gives adequate treatment of the numerical solution of partial differential equations, especially pertaining to problems which involve the use of the Crank-Nicolsen procedure?
  11. S

    Linearity / multilinearity in LDE(linear differential equations)

    hello everyone we demonstrate the linearity in a function by a superposition principle..as in f(x)=y f(x1+x2)=f(x1)+f(x2) but that' the case when we have a single variable as x and if we have two variables then we modify the concept of linearity to multilinearity where f(x,y)=z can never be...
  12. E

    Engineering RC circuit differential equations

    Homework Statement Task is to write differential equation for this circuit. Homework Equations The Attempt at a Solution I'll try to solve the task, but now I want to know, is it possible to use voltage source instead of current source. For example, I can calculate ekvivalent...
  13. T

    Partial Differential Equations

    Solve ##au_{x} + bu_{y} = f(x,y)##, where ##f(x,y)## is a given function. If ##a \neq 0##, write the solution in the form $$u(x,y) = (a^{2} + b^{2})^{\frac{-1}{2}} \int_{L} f ds + g(bx - ay)$$ (from Partial Differential Equations An Introduction, 2nd edition by Walter A. Strauss; pg. 10) I...
  14. W

    Second-Order separable Differential equations

    Homework Statement Solve d2y/dt2 = dx/dt2, if x = 0 and dx/dt = 1 when t = 0 Homework Equations The Attempt at a Solution d2y = dx I'm not exactly sure what to do here the fact that dt2 is under the denominator for both fractions is confusing memaybe its a typo? should it be d2y/dx2 = dx/dt?
  15. S

    Differential Equations: Solving with Two Methods

    Homework Statement Solve each of these differential equations by two different methods. \frac{dy}{dx} = 4(y+1)x^3Homework Equations Integrating factor \rho = \exp (\int(p(x) dx) Linear Equation \frac{dy}{dx} + p(x) y(x) = Q(x) The Attempt at a Solution So I first solved it using...
  16. ThomasMagnus

    How Do Differential Equations Model Learning Performance Over Time?

    Homework Statement Model for learning in the form of a differential equation: \frac{dP}{dt}= k(M-P) Where P(t) measures the performance of someone learning a skill after training time (t), M is the maximum level of performance, and k is a positive constant. Solve this differential...
  17. N

    Differential equations in the schrodinger equation.

    i got a book on differential equations that says a shortcut to solving the general differential equation f'(x)+p(x)f(x)=g(x) is to take the antiderivative of g(x) dx times exp(-p(x) dx times x) to solve for f(x) where dx represents the functions antiderivative. (i kno its supposed to represent...
  18. N

    MHB First Order Differential Equations, given initial value....

    I'm having trouble with this problem... I am almost certain that I have the first part correct which is solving the first order DiffEQ using an integrating factor. I think that I am computing the constant incorrectly. I have followed all steps, including the similar problem given on WileyPlus...
  19. maistral

    Process dynamics modelling for heated tank, differential equations

    I can't seem to model this properly. This isn't an assignment, I'm just curious how this will go, lol. So I have this tank with an incoming feed stream with temperature Ti, and an output stream T. It has a jacket where q would be modified depending on the desired output stream T. So I...
  20. M

    Sinking bucket - differential equations

    Homework Statement Let's have a bucket flowing in water. Now we make a hole underwater. How fast will the bucket sink completely under water? It is a question from course called Ordinary Differential Equation, so I'm supposed to establish an ODE to solve this problem. I understand how to...
  21. E

    Engineering RL circuit differential equations

    Homework Statement Task is to write differential equation for this circuit. E, R1, R2, R3, L are constants. Homework Equations Ul = L di/dt The Attempt at a Solution I guess, we have to use current method for each contour. 1st contour equation: E = U1 + U2 + Ul =...
  22. A

    Ordinary differential equations and BVP

    Solve BVP by separating variables and using eigenfunction expansion method PDE:Ut-Uxx=e-2tsin(pi x/L) U=U(x,t),x(0,L) BC1:U(0,t)=0 BC2:U(L,t)=0 IC:U(x,0)=sin(pi x/L) U(x,t)=X(x)T(t),X''=(lambda) X ,lambda is the separation of parameter. I have calculated the basis functions...
  23. M

    Which easier probabilty or differential equations?

    I'm an engineering student, and in my next semester I want to take one of these 2 courses, differential equations or probability. I'm good in math but I'm taking some hard engineering courses and that's why I'm willing to choose the easiest of these 2 courses. Thank you for your advice.
  24. T

    Is solving differential equations supposed to be this hard?

    Homework Statement $$y''-2y'+5y={ e }^{ x }cos2x$$ Homework Equations The Attempt at a Solution I still haven't completed the question but I just want to know if I'm on the right track. It's becoming ridiculously tiring to just complete 1 question. Is it supposed to be this long?
  25. J

    Solving differential equations with different differential notation?

    Since \frac{dy}{dx} is just considered notation, how can we treat it as an actual fraction when soliving differential equations? Could you, for instance, replace \frac{dy}{dx} with y'(x) in a differential equation and work it out?
  26. B

    Exact Differential Equations of Order n?

    A second order ode Py'' + Qy' + Ry = 0 is exact if there exists a first order ode Ay' + By such that (Ay' + By)' = Ay'' + (A' + B)y' + B'y = Py'' + Qy' + Ry = 0 How can one cast the analysis of this question in terms of exact differential equations? In other words, could somebody...
  27. K

    Solving Differential Equations: Understanding the Steps

    I am looking for help solving these two differential equations: 1. x'=-x 2. x'=x2, x(0)=x0 The solutions are x(t)=e-tx0, and x(t)=x0/(1-x0t). I just don't understand what steps were being done to get those solutions. If someone could point me in the right starting point or show me...
  28. Chris L T521

    MHB Nathan Curtis' Question at Yahoo Answers regarding Differential Equations

    Here is the question: Here is a link to the question: http://answers.yahoo.com/question/index?qid=20130903134631AAuRmGn I have posted a link there to this topic so the OP can find my response.
  29. S

    How to Solve Reducible Exact Differential Equations: Methods Explained

    How can i solve differential equations that are reducible to exact form? please explain each method. thanks
  30. S

    Partial Differential Equations?

    What math subject comes after partial differential equations for physics and electrical engineering majors?
  31. B

    Separable Differential Equations

    I have read that, if you given a differential equation \frac{dy}{dx} = f(x,y), and can write it in the form \frac{dy}{dx} = h(x)g(y), then you can proceed with the following steps: \frac{dy}{g(y)} = h(x)dx integrating G(y) = H(x) + c Why are these steps vaild? I thought that one was not...
  32. V

    Concept to differential equations

    Question: Explain why you cannot solve the ordinary equation? x^2y'' + xy' + (x^2-1)y = 0 My attempt: I don't need to solve it, but just simply state why I can't with just differential equations So my answer is, This differential equation does have a solution, it's just not expressable in...
  33. Astrum

    Differential Equations for Physics

    My weak spot in math is certainly DEs, I find them pretty boring, and fairly unattractive from an aesthetic point of view. I've spent some time with them while studying harmonic motion, and I've gotten OK at them when applied to this particular topic (aside from having trouble with solving...
  34. D

    Entering a Differential Equations

    I'm entering a differential equations course this coming semester. Is there anything I should review in the coming weeks?
  35. L

    Coupled non-linear differential equations

    Homework Statement x'= E - sin x + K sin (y-x) y'= E + sin y + K sin (x-y) E and K >0 Find fixed points for this system of equations Homework Equations This system is the form of coupled oscillators described in Strogatz. θ1'= ω1 + K sin (θ2-θ1) θ2'= ω2 + K sin (θ1-θ2)...
  36. D

    MHB Information on Impulse Differential Equations

    I read a dumb article on a UK mathematician modeling a zombie take over with impulse differential equations. Unbeknownst to me, were impulse DE and the fact that they are used in orbital mechanics. I am unable to find much on this topic tough. I believe it is relative new (1990). Does...
  37. D

    Fundamental laws and differential equations

    why are many fundamental laws of nature formulated in the form of differential equations?
  38. M

    Differential equations of forced oscillation and resonance

    How do I derive A? As you can see in the attachment, I tried to substitute x and expand the equation but I got stuck. How do I get rid of the δ and cos and sin to get the result in the end? Please help!
  39. S

    Differential Equations - Power Series problem with e^t

    Homework Statement The problem is to solve: y''+ty'+e^{t}y=0, y(0)=0 and y'(0)=-1 Homework Equations The Attempt at a Solution My main issue is the following: I normally find the recursion relation, and then factor out the t^{whatever} and I know that the coefficient to this...
  40. V

    Solve these differential equations by converting to Clairaut's form

    The question comprises of three subparts which need to be converted to Clairaut's form and then solved : (a) x p2 - 2yp + x + 2y = 0 (b) x2 p2 + yp (2x + y) + y2 = 0 (c) (x2+y2)(1+p)2-2(x+y)(1+p)(x+yp)+(x+yp)2=0 Note : p = dy/dx I understand that Reducing to Clairaut's form...
  41. S

    Defining functions in terms of differential equations

    I have set myself the task of teaching my Freshman in high school brother Calculus, and today while reviewing some topics I saw something I didn't see before. To start out, I let y = ln[x] => x = e^y Obviously, we know that y' = 1/x = e^-y So, I "discovered" that one can define the...
  42. Fernando Revilla

    MHB Differential Equations (particular solutions)

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  43. M

    Differential Equations Question

    Can anyone tell me how the book arrived at the portion that I underlined in the paint document?
  44. J

    Differential Equations old and the new

    Homework Statement Given: y''[t] + 25 y[t] = 0 I know that the solution to this DE is of the form: y[t] = K1 E^(-5 i t) + K2 ​​E ^(5 i t) I get that, that makes sense to me, however when I look in old DE books I see the solution to the same problem written as: C1 Cos[5 t] + C2...
  45. maistral

    Differential equations for series variable volume reactions

    How do you make a differential equation for such? Say for example. I have two reactions in series, A → R and R → S going in a gas-phase reaction. If I'm correct, the ODE for the conversion of A is dXA/dt = kA*[(NAo/vo)*(1-XA)/(1+δA*YAo*XA)]*[(vo/NAo)*(1+δA*YAo*XA)]. I don't know now...
  46. P

    2nd order nonhomogeneos differential equations with initial conditions

    Homework Statement The problem states d^2y/dt^2 +15y= cost4t + 2sin t initial conditions y(0)=y'(0)=0 Homework Equations The Attempt at a Solution All I have is this r^2+15=0 making r(+-)=√15 and making yh= C1cos√15+C2√15 the next part includes solve for...
  47. P

    2nd order nonhomogeneos differential equations with initial conditions

    I have a problem which in involves a second order differential equations with imaginary roots and I can seem to know how to finish the problem. d^2y/dt^2 +15y =cos 4t+2 sin t this is what I got so far r^2+15=0 for the homogeneous part r=+-(√15) Yh=C1cos√15+C2sin√15 now is...
  48. G

    Differential Equations behavior for large t?

    dy/dt = 2 - 2ty y(0) = 1 I am not asked to solve this (I know it's not easy to solve), but what I am asked is, "for large values of t is the solution y(t) greater than, less than, or equal to 1/t"? I would think less than because 1/e^(t^2) converges faster than 1/t, but at the same...
  49. Q

    Solving Non-Homogeneous Differential Equations with Two y' Terms

    Homework Statement I have to find the differential of (y-xy')^2=x^2+y^2.Now,I have solved hom. equations but this is different because there are two y'. I know how to prove that it is a hom. equation of degree zero, so we can skip that, but how to solve this? Some hints would be highly...
  50. F

    Deriving differential equations for free rotation

    I was asked to formulate the equations governing the rotation of a body moving without any external moments acting about its centre of mass in terms of a coupled system of first order, nonlinear differential equations. I decided to go with the Euler equations, and I ended up with this...
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